MHB A family of functions where each member is its own inverse?

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A family of linear functions where each member is its own inverse can be defined by the property that their slopes must equal one or negative one. When expressed in the form y=ax+b, the inverse function y=a'x+b' will only match the original function if the conditions a=a' and b=b' hold true. This leads to two possible values for the slope a, which must be further constrained by the value of b. Additionally, the inverse function can be visualized as the reflection of the original function across the line y=x, identifying lines that map onto themselves through this reflection. Understanding these properties clarifies the characteristics of such linear functions.
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A family of functions is a set of functions that share one or more properties. ie: The family of quadratics with zeros 1 and 10, or the linear functions with a slope of 20.

there is a family of linear functions where each member is its own inverse. What linear property defines the family?

(I don't really know how to start this, though I think i understand what the question is asking. Help would be appreciated!)
 
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Any linear function can be written uniquely as $y=ax+b$ for some constants $a,b$. Find the inverse function $y=a'x+b'$ ($a'$ and $b'$ are expressed through $a$ and $b$). These functions are the same iff $a=a'$ and $b=b'$. The first of these equations will give you two possible values for $a$. Consider each value to see what constraint the second equation gives you on $b$.

A second way: The graph of the inverse function is obtained by reflecting the graph of the original function w.r.t. the line $y=x$. Find lines that are mapped into themselves by this reflection.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...

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